Lemma 4.2 [1]}

Let \(\left(X,\mathcal {F},\nu \right)\)  be a measure space. Let \(F\,:\,X \times \mathbb {R}_+ \rightarrow \mathbb {R}_+\)  be measurable. Assume that
\(\lim _{M\rightarrow \infty } \frac{\log F(x,M)}{M} = f(x)\) 
